Hofstadter Rules and Generalized Dimensions of the Spectrum of Harper's Equation

TL;DR

This paper investigates the fractal properties of the energy spectrum of Harper's equation for irrational flux, revealing oscillatory behavior in fractal dimensions linked to continued fraction expansion and implications for wavepacket dynamics.

Contribution

It combines Hofstadter rules with semiclassical analysis to relate fractal dimensions of the spectrum to the continued fraction expansion of irrational flux, providing new insights into spectral and dynamical properties.

Findings

Maximal fractal dimension oscillates with continued fraction parameter n.

Minimal fractal dimension asymptotically behaves as ln n / n.

Wavepacket moments are sensitive to the parity of n.

Abstract

We consider the Harper model which describes two dimensional Bloch electrons in a magnetic field. For irrational flux through the unit-cell the corresponding energy spectrum is known to be a Cantor set with multifractal properties. In order to relate the maximal and minimal fractal dimension of the spectrum of Harper's equation to the irrational number involved, we combine a refined version of the Hofstadter rules with results from semiclassical analysis and tunneling in phase space. For quadratic irrationals $\omega$ with continued fraction expansion $\omega = [0;\overline{n}]$ the maximal fractal dimension exhibits oscillatory behavior as a function of $n$, which can be explained by the structure of the renormalization flow. The asymptotic behavior of the minimal fractal dimension is given by $\amin \sim {\rm const.} \ln n / n$. As the generalized dimensions can be related to the anomalous diffusion exponents of an initially localized wavepacket, our results imply that the time evolution of high order moments $< r^{q} >, q \to \infty$ is sensible to the parity of $n$.